The fact that libraries are dying off doesn't bother me that much. That's progress. Old things fall into irrelevance when new things take their place. Their obsolescence was inevitable.
What bothers me is that there isn't a viable alternative yet. Book publishers are likely trying to kill off the libraries as fast as possible, for reasons that should be obvious. Like it's been said already, libraries are a useful public utility. The internet is a good replacement for nonfiction books, but only to those that can afford to access it. Fictional works are another matter entirely, as there is no easy access to them outside of a public library. If there was some kind of digital distribution service that was both free to the public and convenient for all citizens, this would be less of a problem. As it is, people too poor to afford books or computers are out of luck.
There will likely come a time when personal computers and internet access are considered basic necessities for living in a developed country, just like cars are now. That time is rapidly approaching, yet the cost to individual users is still relatively high. A likely solution would be the creation of government-funded ISP that would be freely available to all, but a lot of work on our country's telecommunications infrastructure would be required before that would even become a viable option.